(15 points) Consider the problem of multi-item pricing when the products are substitutable and de- mand is given via an MNL choice model. Let N denote the set of all products and P = {p1, ... ,Pm} the set of feasible prices. Let vi, denote the MNL propensity for product i E N at price P; E P. W.1.o.g. let the propensity of the outside option be 1. In the lecture, we claimed (without proof) that the following LP can be used to obtain the optimal prices. max s.t. PjVizyi; LEN, jelm , - 0. Show that the optimal value of this LP is at least as much as the expected revenue when prices are set optimally and each item is offered with at most one price (i.e., optimal solution to the multi-item pricing problem). (15 points) Consider the problem of multi-item pricing when the products are substitutable and de- mand is given via an MNL choice model. Let N denote the set of all products and P = {p1, ... ,Pm} the set of feasible prices. Let vi, denote the MNL propensity for product i E N at price P; E P. W.1.o.g. let the propensity of the outside option be 1. In the lecture, we claimed (without proof) that the following LP can be used to obtain the optimal prices. max s.t. PjVizyi; LEN, jelm , - 0. Show that the optimal value of this LP is at least as much as the expected revenue when prices are set optimally and each item is offered with at most one price (i.e., optimal solution to the multi-item pricing problem)